I actually would recommend a minimum of 3 gallons per betta. They do take advantage of every bit of swimming space that you give them and are much happier with more space. Even if you go by the ridiculously improper 1 gallon per inch rule, they need at least 3 gallons since they get to be about 3 inches. I also would recommend, as someone else did already, that horizontal space is more important than vertical.
Keep in mind that these fish originate in rice-paddy-like conditions where it's not very deep, but there is a lot of water to swim in (think acres/miles). While they like slow-moving, shallow water, they do need plenty of room otherwise. I would actually try to get a long tank or custom-build a long tank that may only be 18" tall but is quite long which can be divided. As someone already said, opaque or at least semi-opaque panels would be best since most bettas will challenge each other and the constant threat would stress them.
Btw, the idea is pretty cool. I like it.